Question

What is the limit of `(x^9 - 1) / (x^5 - 1)` as x approaches 1?

Answer 1

`9/5`

Explanation:

General background

For any positive integer `n`, we can see that `1` is a zero of the polynomial `x^n-1`.

By the Factor Theorem, `x-1` is a factor of any polynomial of the form `x^n-1`.

Do the division or use trial and error, or learn the result:

`x^n-1 = (x-1)(x^(n-1)+x^(n-2)+x^(n-3)+ * * * +x+1)`

This question

`lim_(xrarr1)(x^9-1)/(x^5-1)` has indeterminate initial form (it has form `0/0`). Therefore `1` is a zero and `x-1` a factor of both the numerator and the denominator.

`lim_(xrarr1)(x^9-1)/(x^5-1) = lim_(xrarr1)(cancel((x-1))(x^8+x^7+x^6+* * * +x+1))/(cancel((x-1))(x^5+x^6+x^4+* * * +x+1)`

`= overbrace(1+1+1+ 1+ * * * +1)^"9 terms"/underbrace(1+1+1+ * * * +1)_"5 terms"`

`= 9/5`